Incremental Approximate Maximum Flow via Residual Graph Sparsification
Gramoz Goranci, Monika Henzinger, Harald Räcke, A. R. Sricharan
TL;DR
The paper addresses maintaining a $(1-\epsilon)$-approximate $s$-$t$ max flow under edge insertions in undirected, uncapacitated graphs. It develops a phase-based incremental algorithm that leverages residual-graph sparsification, Nagamochi–Ibaraki indices for stable sampling, and a balanced directed sparsification framework to preserve augmenting paths with high probability. The main contributions are (i) an incremental sparsifier compatible with approximate max flows, (ii) an extension of cut sparsification to balanced directed graphs, and (iii) an efficient NI-index–based sampling mechanism with near-linear overhead, culminating in a total update time of $\tilde{O}(m\log(n)\alpha(n) + nF^*\log^3(n)/\epsilon)$. This yields polylogarithmic amortized update times for dense graphs and for graphs with small final max flow $(F^* = \tilde{O}(m/n))$, enabling scalable maintenance of approximate flows in dynamic settings.
Abstract
We give an algorithm that, with high probability, maintains a $(1-ε)$-approximate $s$-$t$ maximum flow in undirected, uncapacitated $n$-vertex graphs undergoing $m$ edge insertions in $\tilde{O}(m+ n F^*/ε)$ total update time, where $F^{*}$ is the maximum flow on the final graph. This is the first algorithm to achieve polylogarithmic amortized update time for dense graphs ($m = Ω(n^2)$), and more generally, for graphs where $F^*= \tilde{O}(m/n)$. At the heart of our incremental algorithm is the residual graph sparsification technique of Karger and Levine [SICOMP '15], originally designed for computing exact maximum flows in the static setting. Our main contributions are (i) showing how to maintain such sparsifiers for approximate maximum flows in the incremental setting and (ii) generalizing the cut sparsification framework of Fung et al. [SICOMP '19] from undirected graphs to balanced directed graphs.